The main application of Feynman path integrals (and the primary motivation behind them) is in Quantum Field Theory - currently this is something standard for physicists, if even the mathematical theory of functional integration is not (yet) rigorous.

My question is: what are the applications of path integrals outside QFT? By "outside QFT" I mean non-QFT physics as well as various branches of mathematics.

7 Answers
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In mathematical finance one is faced with the problem of finding the price for an "option."

An option is a contract between a buyer and a seller that gives the buyer the right but not the obligation to buy or sell a specified asset, the underlying, on or before a specified future date, the option's expiration date, at a given price, the strike price. For example, an option may give the buyer the right but not the obligation to buy a stock at some future date at a price set when the contract is settled.

One method of finding the price of such an option involves path integrals. The price of the underlying asset varies with time between when the contract is settled and the expiration date. The set of all possible paths of the underlying in this time interval is the space over which the path integral is evaluated. The integral over all such paths is taken to determine the average pay off the seller will make to the buyer for the settled strike
price. This average price is then discounted, adjusted for for interest, to arrive at the current value of the option.

Statistical Mechanics:

In statistical mechanics the path integral is used in more-or-less the same manner as it is used in quantum field theory. The main difference being a factor of $i$.

One has a given physical system at a given temperature $T$ with an internal energy $U(\phi)$ dependent upon the configuration $\phi$ of the system. The probability that the system is in a given configuration $\phi$ is proportional to

$e^{-U(\phi)/k_B T}$,

where $k_B$ is a constant called the Boltzmann constant. The path integral is then used to determine the average value of any quantity $A(\phi)$ of physical interest

$\left< A \right> := Z^{-1} \int D \phi A(\phi) e^{-U(\phi)/k_B T}$,

where the integral is taken over all configurations and $Z$, the partition function, is used to properly normalize the answer.

Physically Correct Rendering:

Rendering is a process of generating an image from a model through execution of a computer program.

The model contains various lights and surfaces. The properties of a given surface are described by a material. A material describes how light interacts with the surface. The surface may be mirrored, matte, diffuse or any other number of things. To determine the color of a given pixel in the produced image one must trace all possible paths form the lights of the model to the surface point in question. The path integral is used to implement this process through various techniques such as path tracing, photon mapping, and Metropolis light transport.

Topological Quantum Field Theory:

In topological quantum field theory the path integral is used in the exact same manner as it is used in quantum field theory.

Basically, anywhere one uses Monte Carlo methods one is using the path integral.

Some of your examples are about Wiener integrals (minus sign in the exponent rather than imaginary phase), which are matematically well-defined rather than Feynman path integrals which are successfully defined only in some special cases.
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Zoran SkodaApr 6 '10 at 21:03

I agree. All examples I am aware of outside of QFT exchange the i for a -1. Are you aware of any non-QFT examples that have an i in the exponent?
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Kelly DavisApr 6 '10 at 22:26

I usually think of a path integral as just a very glorified and specific version of a simple and general construction from probability. Namely, a path integral is basically an element of an ordered product of matrices belonging to some semigroup. So under this interpretation, "path integrals" are ubiquitous when this sort of object is being considered--particularly in Markov processes. Every time you're computing a multi-step transition probability, you're doing a path integral, and vice versa.

Background:

In discrete-time Markov processes you take a power of the transition matrix. Each element of it encodes all the ways in which you can get from the initial to the final state in the appropriate number of steps, along with their proper weights. In continuous time it's the same basic idea, but a bit more involved. The idea is covered here for inhomogeneous continuous-time processes in the course of demonstrating a fairly general form of the Dynkin formula.

The transfer matrix is an infinitesimal time evolution operator: i.e., $T = U(\Delta t) = \exp(-i \Delta t \cdot H) = I - i\Delta t \cdot H$, where these equalities are up to $o(\Delta t)$. Since time evolution operators belong to a semigroup, we have after a simple manipulation that

Path integral is NOT in general "related" to stationary phase; rather the stationary phase is an asymptotic method for integrals with rapidly oscillating phase, whose infinite dimensional version (that version is to large extent non-rigorous and underdeveloped mathematically) can be sometimes meaningfully APPLIED to the path integral. This is a path integral version of the WKB approximation of the usual approach to QM (nlab). Approximating variational extrema by path integral is equally OK in certain asymptotic regime.
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Zoran SkodaApr 7 '10 at 0:34

Yes You are right - my mistake and inconsistency - in general ( from point of view of some kind of the definition, for example by means of general propagator composed in time ordered points). You are right. But please, could You give me an example of this approach without such method? Possibly the only one is Gaussian path integral in quantum oscillator. Other ones usually are treated in perturbation given by saddle point method.
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kakazApr 7 '10 at 12:20

One application is to computer graphics. When simulating the effect of lighting a translucent material (see my avatar!) you often need to integrate over all possible paths from the light source to the camera via the material. This is similar to the Feynman integral in quantum mechanics, but note that this is an integral in the domain of classical geometric optics, not quantum field theory.

I believe it was Jerry Tessendorf who pioneered this approach in the graphics world. You may have watched movies with effects rendered using techniques derived from Tessendorf's!

I should add that this is a particular case of what Steve Huntsman describes in his answer.

Witten, I think, deserves much of the credit for getting mathematicians interested in the path integral, with his paper Quantum field theory and the Jones polynomial. In particular, path integrals are closely related to questions about (quantum) groups.

For one direction, namely the perturbative Feynman path integral, you should check out Dror Bar-Natan's thesis and later work.

Also, wasn't it Atiyah who first asked whether there was a physics explanation of the Jones polynomial?
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Kevin H. LinApr 6 '10 at 0:25

For a pretty narrow definition of mathematicians: analysts, operator algebraists, and integrable systems people had been thinking about path integrals in various contexts long before Witten got involved. I'm not saying Witten hasn't been influential, especially in geometry and topology, but path integrals and mathematics didn't meet for the first time in the early 80s.
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userNApr 6 '10 at 1:28

Kevin and AJ both make good points, and I apologize for misrepresenting the history. My only excuse is that the Witten paper is a nice place to start a history of the topics I've been most interested in. (Incidentally, I originally posted only Dror's thesis, and then decided that perhaps I should mention Witten's motivation for it.)
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Theo Johnson-FreydApr 6 '10 at 2:08

Some expansions in deformation theory, Lie theory, study of graph cohomology etc. are Feynman integral like expansions and one can formally define "theories leading to them". See for example articles by Dror Bar-Natan for some of such combinatorial and Lie-theoretic aspects. Kontsevich's own deformation quantization formula is usual quantum mechanics is governed by a theory called Poisson sigma model (this was the intuition behind Kontsevich's formula, though he did not explicitly write it that way, but it was later rediscovered by Cattaneo and Felder).

On the other hand, I find very fascinating Sasha Goncharov's "theory" giving a Feynman diagram expansion giving "correlators" formally like in physics, but in fact consisting of Hodge theoretic information on Kähler manifolds: